The paper examines elementary properties of the restricted three-body problem.
Lagrange's mean value theorem is generalised to multivariate case. Using this result, the Euler-Lagrange equations from the calculus of variations are generalised to functionals of vector functions.
The Lagrangian mechanics is then derived from the Hamilton's principle. The Jacobi energy function is defined and its conservation properties examined. Cyclic coordinates are introduced and their connection to the integrals of motion is derived. The basic mechanisms of the Lagrangian mechanics are illustrated and explained by elementary examples. The covariance of the Lagrangian mechanics is proved.
Finally, the restricted three-body problem is examined by the Lagrangian mechanics. The Lagrange function for the secondary (the body whose mass is negligible comparable to the masses of the other two bodies) is derived. Equations that determine the points of equilibrium are derived. Well known analytical solutions are presented. In the case where one of the primaries is of negligible mass comparable to the other one, solutions are numerically approximated to the first order. Finally, the stability of equilibrium points is examined.
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